So first off my guess is your plunger is blown out. Check out this pic...
https://www.pinballlife.com/williamsbally-plunger-and-link-old-style.html
What happens is the plastic piece either cracks completely, or the whole gets blown out creating the slop you're showing in the video.

So a new plunger might fix that part; but a lot of people like to get a flipper rebuild kit. Sometimes it's overkill for a little problem...then they'll also tell you to replace all Mechs are the same time then you're buying 3 of those bad boys.

I can see the slop in your video; the meeting of the plastic plunger & pawl should not have that much play in it.

Edit:Look at 4:25 of that video vs 4 seconds of your first video. Look at your pawl moving around freely and your plunger not moving at all!

Looks like two different 'slops' going on there in the video - the hole (or both holes) in the plastic plunger link is blown out, and the nylon flipper bushing (that the flipper shaft runs thru) is probably badly cracked/broken/worn out causing the wobbling of the end of the flipper shaft.

Agree with all of this. Way too much slop in the link and likely broken bushing (under pawl). Also, the first video shows the EOS switch contacts aren't nearly aligned properly. High power EOS contacts carry all the power for the flippers and need to be perfectly aligned. Rather than trying to adjust that switch properly, I would highly recommend buying a complete rebuild kit for at least the lower flippers. Plungers and links, pawls, coil stops, bushings, EOS switches and return springs. Pinball Life sells rebuild kits.

Just a quick tip, don't upgrade the upper flipper on BK2K to the WPC style as described in Vids guide. When you launch the ball it will hit the tab that holds the flipper return spring (as I found out the hard way the other day).

No. Some people just like to shotgun whole mechs out; not really needed IMO.

If it was me, I'd do the following.

1. Open up pin and raise the playfield to a it's full position.
2. Take a photo of the bottom right flipper, one that you think is working perfectly.
3. Very carefully compare the photos, taking close note of what is different between the two, especially around the leaf switches. Do they look the same? Adjust the top one bending it with a pair of pliers to make it look as close to the bottom as possible.
4. Take a crisp $1 bill, press it between the contacts on that flipper assembly and pull through to clean it.
5. You mentioned the flipper was slipping when you activate it while holding the flipper down. This should not happen. Take a wrench and tighten the nut on the left while either using a pair of pliers to hold the black screw on the right (you have it circled in blue in your photo) or (preferablly) use an allen wrench to tighten the screw on the right. These two combined will "hold" the shaft holding the flipper tighter and keep it from slipping when hit.
6. Before testing it, check by pushing the flipper manually to make sure it turns freely without scraping the playfield. If it does, the flipper needs to be raised a credit cards thickness above the playfield. Failing to do this can result in playfiled art rubbing off.
7. If these items combined do not fix your issue I'd recommend a system 11 flipper rebuild kit. If you buy a pair I'd recommend using them on the right flippers as they are the only ones that hit ramps.

Good luck!

You don't have to replace the entire mech, you can just get a rebuild kit that will have all the parts you need: https://www.pinballlife.com/williamsbally-flipper-rebuild-kit-021988-to-081991.html

Plus you'll need a new bushing: https://www.pinballlife.com/williamsbally-nylon-flipper-bushing.html

https://pinside.com/pinball/forum/topic/vids-guide-to-upgradingrebuilding-flippers

It’s missing the plate that holds the assembly to playfield. Most likely not worn out but if you buy the complete assembly you have taken a lot of the work out of rebuilding the flippers. I guess it’s a question of time vs money. Save the old parts you may need some for the lower flippers as well. However if your going to own pins learning how to rebuild flippers is a fundamental skill if you don’t want to call a tech to your home every few months. Vids guide is highly recommended reading. Good luck bk2k is an awesome game miss mine it’s a game where I learned how to nudge effectively. When they work like they should it’s one of the best system 11 games.
Regards
Al